2. Análise Exploratória de Dados (AED)

A Análise Exploratória de Dados (AED) é uma etapa fundamental na ciência de dados, utilizada para entender a estrutura dos dados, identificar padrões, detectar anomalias e gerar insights iniciais antes da modelagem. Neste capítulo, exploraremos os diferentes tipos de dados e como visualizá-los de forma eficiente. Veremos diversos tipos de gráficos, como histogramas, boxplots, scatter plots e heatmaps, utilizando bibliotecas como Pandas, Matplotlib e Seaborn e a principal usada nesse material, o Plotly. A visualização e análise exploratória por si só, nos torna capazes de extrair informações de grande importância para orientar a tomada de decisões e auxilia fortemente na construção de modelos preditivos.

Devemos primeiramente iterar com os diversos tipos de dados presentes nos conjuntos de dados para compreender sua estrutura e como cada variável pode impactar a análise. Identificar se os dados são numéricos, categóricos ou binários é essencial para escolher os métodos estatísticos e gráficos mais adequados.

Abaixo será carregado um dataset para estudo.

2.1. Biblioteca

Importação dos pacotes usados

import pandas as pd 
import numpy as np
import random
import numpy as np    
import plotly.express as px 
import plotly.io as pio  
import plotly.graph_objs as go

2.2. Pacotes Essenciais: Pandas e Numpy

Antes de começarmos a análise de dados e visualizações, é importante entender as bibliotecas que usaremos ao longo deste trabalho. Entre as mais importantes estão o Pandas e o Numpy, que serão fundamentais para o processamento e análise de dados. Vamos ver brevemente o que cada uma delas faz:

2.2.1. Pandas

O Pandas é uma biblioteca poderosa para manipulação e análise de dados. Ela oferece estruturas de dados como o DataFrame, que facilita a leitura, a transformação e o resumo de grandes conjuntos de dados de forma eficiente. O Pandas permite importar dados de diversos formatos (CSV, Excel, SQL, etc.), realizar operações como filtragem, agrupamento, e transformações, e também facilita a análise exploratória de dados (AED).

Exemplo de uso:

  • Leitura de arquivos CSV

  • Manipulação de dados em tabelas

  • Análise de dados e criação de resumos estatísticos

2.2.2. Numpy

O Numpy é uma biblioteca fundamental para computação científica em Python. Ela fornece suporte para arrays multidimensionais, que são usados para representar e operar com grandes conjuntos de dados numéricos de maneira rápida e eficiente. Além disso, o Numpy contém funções matemáticas avançadas e operações vetorizadas que ajudam a otimizar o desempenho em cálculos matemáticos.

Exemplo de uso:

  • Criação de arrays e matrizes numéricas

  • Operações matemáticas rápidas em grandes volumes de dados

  • Manipulação de dados para modelagem estatística e aprendizado de máquina

2.2.3. Importância dos Pacotes

Durante este Jupyter-Book, o Pandas será utilizado para carregar, limpar e manipular os dados. Já o Numpy será a base para cálculos matemáticos e operações de alto desempenho. Ambas as bibliotecas são essenciais, pois fornecem a estrutura e a velocidade necessárias para realizar análise exploratória de dados (AED), visualizações gráficas e até modelagem preditiva.

Esses pacotes são fundamentais para trabalharmos de forma eficiente e eficaz com os dados ao longo de todo o conteúdo. Vamos ver como podemos usar essas ferramentas nos próximos capítulos para explorar, visualizar e entender melhor os dados.

2.3. Base de dados

dataset = pd.read_csv('data/census.csv')

dataset.head()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income
0 39 State-gov 77516 Bachelors 13 Never-married Adm-clerical Not-in-family White Male 2174 0 40 United-States <=50K
1 50 Self-emp-not-inc 83311 Bachelors 13 Married-civ-spouse Exec-managerial Husband White Male 0 0 13 United-States <=50K
2 38 Private 215646 HS-grad 9 Divorced Handlers-cleaners Not-in-family White Male 0 0 40 United-States <=50K
3 53 Private 234721 11th 7 Married-civ-spouse Handlers-cleaners Husband Black Male 0 0 40 United-States <=50K
4 28 Private 338409 Bachelors 13 Married-civ-spouse Prof-specialty Wife Black Female 0 0 40 Cuba <=50K

Podemos observar diferentes tipos de dados:

  1. Variáveis Numéricas São aquelas que representam valores quantitativos, permitindo operações matemáticas diretas, como média e desvio padrão. No dataset, temos as seguintes variáveis numéricas:

    • age: Idade do indivíduo.

    • final-weight: Peso final atribuído a cada indivíduo no censo.

    • education-num: Número de anos de educação formal.

    • capital-gain: Ganhos de capital.

    • capital-loos: Perdas de capital.

    • hour-per-week: Número de horas trabalhadas por semana.

  2. Variáveis Categóricas São variáveis que representam grupos ou categorias, sem uma ordem numérica natural. No dataset, temos:

workclass: Tipo de emprego (ex: “State-gov”, “Self-emp-not-inc”). education: Nível educacional (ex: “Bachelors”, “Masters”). marital-status: Estado civil (ex: “Never-married”, “Married-civ-spouse”). occupation: Tipo de ocupação (ex: “Adm-clerical”, “Exec-managerial”). relationship: Papel familiar (ex: “Husband”, “Not-in-family”). race: Raça do indivíduo (ex: “White”, “Black”). sex: Gênero (“Male” ou “Female”). native-country: País de origem (ex: “United-States”, “Mexico”).

  1. Variável Binária Variáveis binárias assumem apenas dois valores distintos. No dataset, a variável income se enquadra nessa categoria:

income: Indica se o indivíduo ganha mais ou menos de 50K por ano (<=50K ou >50K).

Para verificar os tipos de dados no Pandas, utilizamos:

print(dataset.dtypes)

age                int64
workclass         object
final-weight       int64
education         object
education-num      int64
marital-status    object
occupation        object
relationship      object
race              object
sex               object
capital-gain       int64
capital-loos       int64
hour-per-week      int64
native-country    object
income            object
dtype: object

Podemos verificar a distribuição da variável binária com:

print(dataset['income'].value_counts())

 <=50K    24720
 >50K      7841
Name: income, dtype: int64

2.4. Visualização Gráfica

A Análise Exploratória de Dados (AED) é uma abordagem fundamental para entender conjuntos de dados, permitindo identificar padrões, outliers e tendências. Além de métodos estatísticos, a visualização gráfica é uma ferramenta essencial para resumir e interpretar as características principais dos dados de forma intuitiva.

Neste capítulo, exploraremos diferentes técnicas de visualização para variáveis numéricas e categóricas, utilizando gráficos que facilitam a interpretação dos dados. O objetivo é fornecer uma visão clara sobre a distribuição, correlação e comportamento das variáveis do nosso dataset.

2.4.1. Uso do Plotly para Visualizações Interativas

Para a construção dos gráficos, optamos pelo uso do Plotly, pois suas visualizações interativas oferecem uma experiência mais rica ao explorar os dados. Quando este livro for visualizado online, o leitor poderá interagir com os gráficos, analisando detalhes específicos ao passar o cursor sobre os elementos ou ampliando regiões de interesse.

Abaixo, um exemplo simples de um histograma interativo para visualizar a distribuição da idade dos indivíduos no dataset:

from plotly.offline import plot
from IPython.display import display, HTML
pio.renderers.default = 'jupyterlab'

def plotly_plot(fig):
    plot(fig, filename='temp-plot.html', auto_open=False)
    display(HTML('temp-plot.html'))

fig = px.histogram(dataset, x="age", nbins=30, title="Distribuição de Idade")
# Gerar e exibir o gráfico de forma interativa
fig.show()
plotly_plot(fig)

Nota: Defini uma função chamada plotly_plot que recebe um objeto de figura e gera um gráfico interativo. O gráfico é salvo em um arquivo HTML temporário e exibido no notebook. Isso se deve pelo fato de que renderizar figuras no jupyterbook ainda é um desafio. Assim, todas as figuras plotly geradas serão mostradas localmente com fig.show e serão plotadas para o Jupyterbook com plotly_plot

2.4.2. Criando um tema personalizado para o plotly

Existem temas pré-definidos para o plotly porém aqui optamos por customizar um tema a fim de deixar os gráficos mais agradáveis visualmente.

custom_template = pio.templates["simple_white"]
 
custom_template.layout.title = {   'y': 0.95 ,
    'font': {'color': 'black', 'family': 'HelveticaNeue-CondensedBold, Helvetica, Sans-serif', 'size': 25}
}
 
custom_template.layout.font =  ({'family': 'Helvetica Neue, Helvetica, Sans-serif', 'size': 17, 'color': '#000000'})
 
custom_template.layout.hovermode = 'x unified'
 
custom_template.data['bar'] =  ( ({
    'textfont': {'color': '#FFFFFF', 'family': 'Helvetica Neue, Helvetica, Sans-serif', 'size': 20},
    'textposition': 'outside',
    'texttemplate': '%{value:.2s}'
}),)
 
custom_template.layout['xaxis']['showgrid'] = True
custom_template.layout['yaxis']['showgrid'] = True
 
custom_template.layout.autosize = False
 
custom_template.layout.width = 900
custom_template.layout.height = 540
 
custom_template.data.scatter = [go.Scatter(marker=dict(symbol="circle", size=7))]
custom_template.layout.colorway = ['#0000FF', '#FF0000'] + list(px.colors.qualitative.G10)[2:]
custom_template.layout.legend = {"itemsizing": "constant", "itemwidth": 30}
 
pio.templates.default = custom_template

Agora com o novo template:

fig = px.histogram(dataset, x="age", nbins=30, title="Distribuição de Idade", template=custom_template)
fig.update_layout(xaxis_title="Idade", yaxis_title="Contagem") # colocando labels em português
fig.show()
plotly_plot(fig)

Podemos discriminar por sexo:

fig = px.histogram(dataset, x="age", color='sex',nbins=30, title="Distribuição de Idade", template=custom_template)
fig.update_layout(xaxis_title="Idade", yaxis_title="Contagem", legend_title_text='sexo') # colocando labels em português
fig.show()
plotly_plot(fig)

Podemos também fazer gráficos de dispersão (Scatter Plots) para visualizar relações entre duas variáveis. Exemplo: horas trabalhadas por semana vs. capital ganho.

fig = px.scatter(dataset, x="hour-per-week", y="capital-gain", color="income", opacity=0.7, template=custom_template)
fig.update_layout(title="Relação entre Horas Trabalhadas e Capital Gain")
fig.show()
plotly_plot(fig)

Para explorar a distribuição de uma variável contínua como idade ou salário por nível educacional, um boxplot é ideal:

fig = px.box(dataset, x="education", y="age", color="education", template=custom_template)
fig.update_layout(title="Distribuição de Idade por Nível Educacional")
fig.show()
plotly_plot(fig)

Para gráficos de linha (Line Plots) um bom exemplo é um dataframe que envolva datas, assim podemos visualizar tendências no tempo. Podemos usar o dataset “Diabetes” do sklearn.datasets, que contém informações sobre pacientes diabéticos, incluindo medições ao longo do tempo.

Segue um exemplo de gráfico de linha usando Plotly:

from sklearn.datasets import load_diabetes  
# Carregar o dataset do scikit-learn
diabetes = load_diabetes(as_frame=True)
df_diabetes = diabetes.frame

# Criar um índice temporal fictício (apenas para visualização)
df_diabetes["time"] = pd.date_range(start="2020-01-01", periods=len(df_diabetes), freq="D")

# Criar um gráfico de linha: evolução do IMC ao longo do tempo
fig = px.line(df_diabetes, x="time", y="bmi", title="Evolução do IMC ao Longo do Tempo")

fig.show() 
plotly_plot(fig)

2.5. Medidas de Tendência Central

As Medidas de Tendência Central são estatísticas que resumem um conjunto de dados indicando um valor central ou típico. Elas são fundamentais para entender a distribuição dos dados e são amplamente utilizadas na Análise Exploratória de Dados (AED).

As principais medidas de tendência central são:

2.5.1. Média

A média aritmética é calculada somando todos os valores e dividindo pelo número total de observações. Ela é sensível a outliers e pode ser influenciada por valores extremos.

A fórmula para a média é ( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i ). Em Python, podemos calcular a média com o Pandas:

dataset['age'].mean()

38.58164675532078

2.5.2. Mediana

A mediana é o valor central dos dados quando eles estão ordenados. Se houver um número par de elementos, a mediana será a média dos dois valores centrais.

Ela é mais robusta que a média para distribuições assimétricas ou com outliers.

dataset['age'].median()

37.0

2.5.3. Moda

A moda é o valor que mais se repete no conjunto de dados.

Se houver múltiplos valores com a mesma frequência, pode haver mais de uma moda (distribuição multimodal).

Em Python:

dataset['age'].mode()

0    36
Name: age, dtype: int64

2.5.4. Comparação entre as Medidas de Tendência Central

Cada medida de tendência central possui características que a tornam mais apropriada para diferentes situações. A tabela a seguir resume as vantagens e desvantagens de cada uma:

Medida

Vantagem

Desvantagem

Média

Usa todos os valores da amostra, sendo útil para comparações estatísticas.

Sensível a outliers, podendo distorcer a interpretação dos dados.

Mediana

Resistente a outliers e distribuições assimétricas, representando melhor o valor central em dados não normais.

Não utiliza todos os valores na sua determinação.

Moda

Indica o valor mais comum nos dados, sendo útil para variáveis categóricas.

Pode não ser única (distribuição multimodal) ou não existir em conjuntos sem repetições.

A escolha da medida apropriada depende do contexto da análise. Em distribuições simétricas, a média e a mediana costumam ser próximas. Já em distribuições assimétricas ou com outliers, a mediana é uma melhor representação do centro dos dados.

2.5.5. Média Truncada

A média truncada é uma variação da média aritmética em que uma porcentagem dos valores mais altos e mais baixos do conjunto de dados é removida antes do cálculo. Essa abordagem reduz a influência de outliers e valores extremos, tornando-a uma alternativa robusta à média tradicional.

2.5.5.1. Comparação entre Média Truncada e Mediana

Métrica

Média Truncada

Mediana

Definição

Calculada após remover uma fração dos menores e maiores valores do conjunto.

O valor central da distribuição, que divide os dados em duas partes iguais.

Robustez a Outliers

Menos sensível do que a média tradicional, mas ainda pode ser influenciada por valores extremos remanescentes.

Extremamente robusta, pois considera apenas a posição dos dados e não os valores em si.

Uso Recomendado

Quando há outliers moderados, mas a média ainda deve refletir a maioria dos dados.

Quando há outliers extremos ou distribuições altamente assimétricas.

Informação Utilizada

Usa a maioria dos valores, exceto a parte truncada.

Considera apenas a posição relativa dos valores.

2.5.5.2. Exemplo em Python

Podemos calcular a média truncada utilizando a função trim_mean da biblioteca scipy.stats:

from scipy.stats import trim_mean

dados = dataset['age']
media_truncada = trim_mean(dados, proportiontocut=0.1)  # Remove 10% das extremidades
mediana = np.median(dados)

print(f"Média Truncada: {media_truncada}")
print(f"Mediana: {mediana}")

Média Truncada: 37.68981534799801
Mediana: 37.0

No próximo tópico, exploraremos Medidas de Dispersão, que ajudam a entender a variabilidade dos dados.

3. Amostragem e Distribuições

A amostragem é um conceito fundamental em estatística e ciência de dados, pois permite que trabalhemos com subconjuntos de dados representativos de uma população maior. De [Bruce et al., 2020], sabemos que a escolha adequada de uma amostra é crucial para garantir que as inferências feitas a partir dos dados sejam válidas e generalizáveis. Existem diferentes técnicas de amostragem, como a amostragem aleatória simples, amostragem estratificada e amostragem sistemática, cada uma com suas vantagens e aplicações específicas.

Além disso, as distribuições desempenham um papel central na análise estatística. Uma distribuição descreve como os valores de uma variável estão dispersos ou concentrados. Conforme destacado por [Bruce et al., 2020], entender a distribuição dos dados é essencial para aplicar métodos estatísticos corretamente. Por exemplo, a distribuição normal (ou gaussiana) é amplamente utilizada devido às suas propriedades matemáticas e à sua frequência em fenômenos naturais. Outras distribuições, como a binomial, Poisson e exponencial, também são importantes, dependendo do contexto dos dados.

A relação entre amostragem e distribuições é evidente quando consideramos o Teorema Central do Limite (serpá discutido em detalhes posteriormente). Esse teorema afirma que, à medida que o tamanho da amostra aumenta, a distribuição das médias amostrais tende a uma distribuição normal, independentemente da forma da distribuição da população original. Esse princípio é fundamental para muitas técnicas estatísticas, como a construção de intervalos de confiança e testes de hipóteses [Bruce et al., 2020].

dataset.shape

(32561, 15)

dataset.head(10)
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income
0 39 State-gov 77516 Bachelors 13 Never-married Adm-clerical Not-in-family White Male 2174 0 40 United-States <=50K
1 50 Self-emp-not-inc 83311 Bachelors 13 Married-civ-spouse Exec-managerial Husband White Male 0 0 13 United-States <=50K
2 38 Private 215646 HS-grad 9 Divorced Handlers-cleaners Not-in-family White Male 0 0 40 United-States <=50K
3 53 Private 234721 11th 7 Married-civ-spouse Handlers-cleaners Husband Black Male 0 0 40 United-States <=50K
4 28 Private 338409 Bachelors 13 Married-civ-spouse Prof-specialty Wife Black Female 0 0 40 Cuba <=50K
5 37 Private 284582 Masters 14 Married-civ-spouse Exec-managerial Wife White Female 0 0 40 United-States <=50K
6 49 Private 160187 9th 5 Married-spouse-absent Other-service Not-in-family Black Female 0 0 16 Jamaica <=50K
7 52 Self-emp-not-inc 209642 HS-grad 9 Married-civ-spouse Exec-managerial Husband White Male 0 0 45 United-States >50K
8 31 Private 45781 Masters 14 Never-married Prof-specialty Not-in-family White Female 14084 0 50 United-States >50K
9 42 Private 159449 Bachelors 13 Married-civ-spouse Exec-managerial Husband White Male 5178 0 40 United-States >50K 
dataset.tail(5)
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income
32556 27 Private 257302 Assoc-acdm 12 Married-civ-spouse Tech-support Wife White Female 0 0 38 United-States <=50K
32557 40 Private 154374 HS-grad 9 Married-civ-spouse Machine-op-inspct Husband White Male 0 0 40 United-States >50K
32558 58 Private 151910 HS-grad 9 Widowed Adm-clerical Unmarried White Female 0 0 40 United-States <=50K
32559 22 Private 201490 HS-grad 9 Never-married Adm-clerical Own-child White Male 0 0 20 United-States <=50K
32560 52 Self-emp-inc 287927 HS-grad 9 Married-civ-spouse Exec-managerial Wife White Female 15024 0 40 United-States >50K

3.1. Amostragem aleatória simples (AAS)

A Amostragem Aleatória Simples (AAS) é um método estatístico de seleção de amostras onde cada elemento da população tem a mesma probabilidade de ser escolhido. Essa técnica garante que a amostra seja representativa, reduzindo vieses na seleção dos dados.

Para realizar uma amostragem aleatória simples, pode-se:

  1. Numerar os elementos da população.

  2. Selecionar aleatoriamente os indivíduos usando um sorteio, gerador de números aleatórios ou outra técnica estatística.

Exemplo: Se temos uma população de 1.000 pessoas e queremos selecionar uma amostra de 100 indivíduos, cada um deles tem 1/1.000 de chance de ser escolhido.

3.1.1. Vantagens da AAS:

Simplicidade e facilidade de implementação

  • O método é intuitivo e pode ser aplicado sem necessidade de cálculos complexos.

Baixo viés na seleção da amostra

  • Como todos os elementos têm a mesma probabilidade de serem escolhidos, a amostra tende a ser representativa da população.

Fundamento estatístico sólido

  • A análise inferencial baseada na AAS é bem desenvolvida, permitindo a aplicação de testes estatísticos e estimativas confiáveis.

3.1.2. Desvantagens da AAS:

Pode não ser viável para populações grandes

  • Quando a população é muito grande, listar todos os indivíduos e realizar uma seleção aleatória pode ser impraticável.

Não garante total representatividade para subgrupos

  • Se há subgrupos distintos dentro da população (exemplo: diferentes faixas etárias, regiões geográficas), a AAS pode gerar amostras que não refletem adequadamente esses grupos.

Pode ser ineficiente em termos de custo e tempo

  • Se os dados estão dispersos geograficamente, coletar amostras aleatórias pode ser caro e trabalhoso.

3.1.3. Limitações da AAS:

Necessidade de uma lista completa da população

  • Para aplicar a AAS corretamente, é necessário ter acesso à totalidade da população, o que pode ser inviável em muitos cenários.

Problemas com variabilidade da amostra

  • Como a seleção é aleatória, pode haver flutuações nas características da amostra, gerando estimativas menos precisas em comparação com métodos estratificados.

Não funciona bem para populações heterogêneas

  • Se a população tem grande variabilidade interna, uma AAS pode não capturar todas as características importantes, sendo necessário adotar técnicas como Amostragem Estratificada ou Amostragem por Conglomerados.

A seguir vamos trabalhar na base dataset para gerar amostras.

df_amostra_aleatoria_simples = dataset.sample(n = 100, random_state = 1)

df_amostra_aleatoria_simples.shape

(100, 15)

df_amostra_aleatoria_simples.head()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income
9646 62 Self-emp-not-inc 26911 7th-8th 4 Widowed Other-service Not-in-family White Female 0 0 66 United-States <=50K
709 18 Private 208103 11th 7 Never-married Other-service Other-relative White Male 0 0 25 United-States <=50K
7385 25 Private 102476 Bachelors 13 Never-married Farming-fishing Own-child White Male 27828 0 50 United-States >50K
16671 33 Private 511517 HS-grad 9 Married-civ-spouse Prof-specialty Husband White Male 0 0 40 United-States <=50K
21932 36 Private 292570 11th 7 Never-married Machine-op-inspct Unmarried White Female 0 0 40 United-States <=50K

Definindo uma função de AAS,

def amostragem_aleatoria_simples(dataset, amostras):
  return dataset.sample(n = amostras, random_state=1)

df_amostra_aleatoria_simples = amostragem_aleatoria_simples(dataset, 100)
df_amostra_aleatoria_simples.shape

(100, 15)

df_amostra_aleatoria_simples.head()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income
9646 62 Self-emp-not-inc 26911 7th-8th 4 Widowed Other-service Not-in-family White Female 0 0 66 United-States <=50K
709 18 Private 208103 11th 7 Never-married Other-service Other-relative White Male 0 0 25 United-States <=50K
7385 25 Private 102476 Bachelors 13 Never-married Farming-fishing Own-child White Male 27828 0 50 United-States >50K
16671 33 Private 511517 HS-grad 9 Married-civ-spouse Prof-specialty Husband White Male 0 0 40 United-States <=50K
21932 36 Private 292570 11th 7 Never-married Machine-op-inspct Unmarried White Female 0 0 40 United-States <=50K

3.2. Amostragem sistemática

A Amostragem Sistemática é um método de amostragem probabilística onde os elementos da população são selecionados em intervalos regulares a partir de uma lista ordenada. Diferente da Amostragem Aleatória Simples (AAS), em que cada elemento é escolhido independentemente, aqui definimos um intervalo de seleção \( k \) e selecionamos os elementos seguindo esse padrão.

3.2.1. Passos para aplicar a Amostragem Sistemática**

  1. Determinar o tamanho da população \( N \)** e o tamanho da amostra desejada \( n \).

  2. Calcular o intervalo de seleção \( k \)**:

    \[ k = \frac{N}{n} \]

  3. Escolher aleatoriamente um ponto inicial \( r \)** entre 1 e \( k \).

  4. Selecionar os próximos elementos** somando \( k \) ao índice inicial \( r \).

3.2.1.1. Exemplo prático*

Se temos uma população de 1.000 indivíduos e queremos uma amostra de 100, calculamos:

\[ k = \frac{1000}{100} = 10 \]

Se o número aleatório inicial for 7, os elementos selecionados serão: 7, 17, 27, 37, …, 997.

3.2.2. Vantagens da Amostragem Sistemática

Fácil de aplicar e rápida

  • Exige apenas um sorteio inicial, tornando-a mais eficiente do que a Amostragem Aleatória Simples.

Boa cobertura da população

  • Como os elementos são escolhidos em intervalos fixos, a distribuição da amostra tende a ser mais homogênea.

Eficiente para populações grandes e organizadas

  • Útil quando a população está organizada em uma lista ou banco de dados.

3.2.3. Desvantagens da Amostragem Sistemática

Pode gerar viés se houver padrões na população

  • Se a lista seguir um padrão cíclico que coincida com o intervalo \( k \), a amostra pode não ser representativa.

  • Exemplo: Em uma fábrica, se os funcionários estiverem organizados por turnos e a seleção seguir um ciclo que sempre cai no mesmo turno, a amostra será enviesada.

**Menos flexível do que a Amostragem Aleatória Simples

  • Depois de definido o intervalo \( k \), a amostra não pode ser ajustada dinamicamente.

3.2.4. Relação com outros métodos de amostragem

3.2.4.1. Amostragem Sistemática vs. Amostragem Estratificada

  • Amostragem Estratificada (que será utilizada adiante) divide a população em grupos homogêneos (estratos) e, dentro de cada estrato, a amostragem pode ser feita de forma aleatória ou sistemática.

  • Aplicação

    • Amostragem Sistemática: Usada quando a população é relativamente homogênea e organizada.

    • Amostragem Estratificada: Usada quando há subgrupos distintos e queremos garantir que cada um esteja representado na amostra.

    Exemplo: Se queremos pesquisar hábitos de consumo em diferentes faixas etárias, a Amostragem Estratificada é melhor, pois cada faixa etária pode ter um comportamento diferente.

3.2.4.2. Amostragem Sistemática vs. Amostragem por Conglomerados

  • Amostragem por Conglomerados divide a população em grupos naturais (conglomerados) e seleciona alguns desses grupos aleatoriamente para análise.

  • Aplicação

    • Amostragem Sistemática: Usada quando podemos listar toda a população e queremos distribuir bem os elementos na amostra.

    • Amostragem por Conglomerados: Usada quando não conseguimos listar toda a população, mas podemos dividir em grupos naturais.

    Exemplo: Se queremos pesquisar a qualidade da educação em escolas de uma cidade, não precisamos amostrar alunos individualmente. Podemos sortear algumas escolas (conglomerados) e analisar todos os alunos dessas escolas.

Algumas observações:

dataset.shape

(32561, 15)

len(dataset) // 100

325

random.seed(1)
random.randint(0, 325)

68

68 + 325

393

393 + 325

718

np.arange(68, len(dataset), step = 325)

array([   68,   393,   718,  1043,  1368,  1693,  2018,  2343,  2668,
        2993,  3318,  3643,  3968,  4293,  4618,  4943,  5268,  5593,
        5918,  6243,  6568,  6893,  7218,  7543,  7868,  8193,  8518,
        8843,  9168,  9493,  9818, 10143, 10468, 10793, 11118, 11443,
       11768, 12093, 12418, 12743, 13068, 13393, 13718, 14043, 14368,
       14693, 15018, 15343, 15668, 15993, 16318, 16643, 16968, 17293,
       17618, 17943, 18268, 18593, 18918, 19243, 19568, 19893, 20218,
       20543, 20868, 21193, 21518, 21843, 22168, 22493, 22818, 23143,
       23468, 23793, 24118, 24443, 24768, 25093, 25418, 25743, 26068,
       26393, 26718, 27043, 27368, 27693, 28018, 28343, 28668, 28993,
       29318, 29643, 29968, 30293, 30618, 30943, 31268, 31593, 31918,
       32243])

3.2.5. Exemplo prático

Função para amostragem sistemática:

def amostragem_sistematica(dataset, amostras):
    # Calculando o intervalo entre as amostras
    intervalo = len(dataset) // amostras

    # Inicializando a semente aleatória para reprodutibilidade
    random.seed(1)

    # Gerando um índice inicial aleatório dentro do intervalo
    inicio = random.randint(0, intervalo)

    # Criando os índices para a amostragem
    indices = np.arange(inicio, len(dataset), step=intervalo)

    # Selecionando a amostra sistemática
    amostra_sistematica = dataset.iloc[indices]

    return amostra_sistematica

df_amostra_sistematica = amostragem_sistematica(dataset, 100)
df_amostra_sistematica.shape

(100, 15)

df_amostra_sistematica.head()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income
68 49 Self-emp-inc 191681 Some-college 10 Married-civ-spouse Exec-managerial Husband White Male 0 0 50 United-States >50K
393 34 State-gov 98101 Bachelors 13 Married-civ-spouse Exec-managerial Husband White Male 7688 0 45 ? >50K
718 22 Private 214399 Some-college 10 Never-married Sales Own-child White Female 0 0 15 United-States <=50K
1043 44 Private 167005 Bachelors 13 Married-civ-spouse Exec-managerial Husband White Male 7688 0 60 United-States >50K
1368 52 Private 152234 HS-grad 9 Married-civ-spouse Exec-managerial Husband Asian-Pac-Islander Male 99999 0 40 Japan >50K

Portanto, a amostra selecionada a partir do dataset original contém 100 registros, e cada registro possui 16 características diferentes. Essas características são as colunas do seu dataset original, que podem incluir variáveis como age, workclass, education, entre outras (como mostrado no cabeçalho do seu dataset).

3.3. Amostragem por grupos

Este tipo de Amostragem simplesmente divide o conjunto de dados de forma sequencial (ou seja, por ordem de entrada) em grupos iguais, sem considerar a variabilidade interna dos grupos. Isso é mais uma divisão estratificada ou simples divisão em grupos. A amostragem não é aleatória e não leva em consideração a estrutura interna dos dados.

len(dataset) / 10

3256.1

# Determina o número de grupos
num_grupos = 10

# Cria uma lista com o número do grupo correspondente para cada linha
grupos = np.repeat(np.arange(num_grupos), len(dataset) // num_grupos + 1)[:len(dataset)]

# Adiciona ao dataset (se necessário)
dataset['grupo'] = grupos

print(grupos)

[0 0 0 ... 9 9 9]

np.unique(grupos, return_counts=True)

(array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]),
 array([3257, 3257, 3257, 3257, 3257, 3257, 3257, 3257, 3257, 3248]))

np.shape(grupos), dataset.shape

((32561,), (32561, 16))

dataset.head()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income grupo
0 39 State-gov 77516 Bachelors 13 Never-married Adm-clerical Not-in-family White Male 2174 0 40 United-States <=50K 0
1 50 Self-emp-not-inc 83311 Bachelors 13 Married-civ-spouse Exec-managerial Husband White Male 0 0 13 United-States <=50K 0
2 38 Private 215646 HS-grad 9 Divorced Handlers-cleaners Not-in-family White Male 0 0 40 United-States <=50K 0
3 53 Private 234721 11th 7 Married-civ-spouse Handlers-cleaners Husband Black Male 0 0 40 United-States <=50K 0
4 28 Private 338409 Bachelors 13 Married-civ-spouse Prof-specialty Wife Black Female 0 0 40 Cuba <=50K 0 
dataset.tail()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income grupo
32556 27 Private 257302 Assoc-acdm 12 Married-civ-spouse Tech-support Wife White Female 0 0 38 United-States <=50K 9
32557 40 Private 154374 HS-grad 9 Married-civ-spouse Machine-op-inspct Husband White Male 0 0 40 United-States >50K 9
32558 58 Private 151910 HS-grad 9 Widowed Adm-clerical Unmarried White Female 0 0 40 United-States <=50K 9
32559 22 Private 201490 HS-grad 9 Never-married Adm-clerical Own-child White Male 0 0 20 United-States <=50K 9
32560 52 Self-emp-inc 287927 HS-grad 9 Married-civ-spouse Exec-managerial Wife White Female 15024 0 40 United-States >50K 9 
def amostragem_agrupamento(dataset, numero_grupos):
    # Calcular o intervalo para dividir os dados em grupos
    intervalo = len(dataset) // numero_grupos
    
    # Gerar os grupos com base no intervalo
    dataset['grupo'] = [i // intervalo for i in range(len(dataset))]
    
    # Garantir que os grupos sejam numerados corretamente caso o número de registros não seja exato
    dataset['grupo'] = dataset['grupo'].apply(lambda x: min(x, numero_grupos - 1))

    # Semente aleatória para garantir reprodutibilidade
    random.seed(1)
    
    # Selecionar aleatoriamente um grupo
    grupo_selecionado = random.randint(0, numero_grupos - 1)
    
    # Retornar o subset com os dados do grupo selecionado
    return dataset[dataset['grupo'] == grupo_selecionado]

df_amostra_agrupamento = amostragem_agrupamento(dataset, 325)
df_amostra_agrupamento.shape, df_amostra_agrupamento['grupo'].value_counts()

((100, 16),
 68    100
 Name: grupo, dtype: int64)

df_amostra_agrupamento.head()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income grupo
6800 52 Federal-gov 424012 Some-college 10 Divorced Exec-managerial Not-in-family White Female 0 0 40 United-States <=50K 68
6801 29 Private 178469 HS-grad 9 Never-married Sales Not-in-family White Female 0 0 40 United-States <=50K 68
6802 73 Self-emp-inc 92886 10th 6 Widowed Sales Unmarried White Female 0 0 40 Canada <=50K 68
6803 38 Self-emp-not-inc 214008 HS-grad 9 Married-civ-spouse Prof-specialty Husband White Male 0 0 50 United-States >50K 68
6804 59 Self-emp-not-inc 325732 HS-grad 9 Married-civ-spouse Exec-managerial Husband White Male 0 1902 52 United-States >50K 68

3.4. Amostra estratificada

A amostragem estratificada é um método de amostragem probabilística em que a população é dividida em subgrupos ou “estratos” com base em características específicas, e em seguida, amostras são retiradas de cada um desses estratos. O objetivo principal desse tipo de amostragem é garantir que diferentes subgrupos importantes da população sejam representados de forma adequada na amostra, o que aumenta a precisão das estimativas quando comparado à amostragem aleatória simples.

3.4.1. Definindo a amostra:

  1. Divisão da população em estratos:

    • A população é dividida em subgrupos (estratos) com base em uma ou mais características específicas, como idade, gênero, renda, localização geográfica, etc.

    • Cada estrato deve ser mutuamente exclusivo, ou seja, um indivíduo da população só pode pertencer a um único estrato.

  2. Amostragem dentro de cada estrato:

    • Após a divisão da população em estratos, amostras são selecionadas dentro de cada estrato. A seleção pode ser feita de forma aleatória, mas a quantidade de amostras retiradas de cada estrato pode variar dependendo do tipo de amostragem estratificada:

      • Amostragem proporcional: O número de amostras retiradas de cada estrato é proporcional ao tamanho de cada estrato na população total. Por exemplo, se 30% da população está em um estrato, 30% da amostra será retirada desse estrato.

      • Amostragem igual: Uma quantidade igual de amostras é retirada de cada estrato, independentemente do tamanho relativo dos estratos na população total. Isso pode ser útil se você quiser garantir que grupos menores sejam igualmente representados na amostra.

      • Amostragem ótima: A quantidade de amostras retirada de cada estrato é determinada com base na variabilidade dentro de cada estrato e no custo de amostragem, com o objetivo de minimizar o erro de estimação.

  3. Combinação das amostras:

    • Após as amostras serem retiradas de cada estrato, elas são combinadas para formar a amostra final que será analisada.

3.4.2. Vantagens da amostragem estratificada:

  • Maior precisão: Como os estratos são grupos homogêneos, a amostragem dentro desses grupos tende a ser mais precisa do que uma amostra aleatória simples, onde a variabilidade dentro da amostra pode ser maior.

  • Representatividade de subgrupos: A amostragem estratificada garante que subgrupos importantes sejam representados adequadamente na amostra, o que pode ser crucial em pesquisas com diferentes populações ou características.

  • Eficiência: Pode ser mais eficiente em termos de custo e tempo, pois concentra a amostragem nos estratos mais relevantes para o estudo.

3.4.3. Exemplo de amostragem estratificada:

Imagine que você está conduzindo uma pesquisa sobre os hábitos de consumo de diferentes faixas etárias em uma população de 1.000 pessoas, onde:

  • 200 são jovens (18-30 anos),

  • 500 são adultos (31-50 anos),

  • 300 são idosos (acima de 50 anos).

Se você quiser uma amostra de 100 pessoas com uma amostragem proporcional, você selecionaria:

  • 20 jovens (10% da população),

  • 50 adultos (50% da população),

  • 30 idosos (30% da população).

Caso você opte por uma amostragem igual, você selecionaria 33 pessoas de cada estrato, independentemente de sua proporção na população.

3.4.4. Considerações:

  • A amostragem estratificada é particularmente útil quando a variabilidade entre os estratos é alta e você quer garantir que a amostra seja representativa desses estratos.

  • É necessário ter conhecimento prévio sobre a distribuição dos estratos na população, o que pode ser uma limitação dependendo dos dados disponíveis.

3.4.5. Aplicações:

  • Pesquisa de mercado: Para garantir que diferentes grupos de consumidores sejam representados adequadamente.

  • Pesquisas sociais e políticas: Para garantir que diferentes segmentos da população (por exemplo, regiões, classes sociais, etnias) sejam adequadamente refletidos.

  • Estudos médicos: Para garantir que diferentes grupos de pacientes (por exemplo, com diferentes condições de saúde) sejam representados na amostra.

A amostragem estratificada é um método poderoso, principalmente quando se trata de amostras complexas e a precisão é uma prioridade.

from sklearn.model_selection import StratifiedShuffleSplit

dataset['income'].value_counts()

 <=50K    24720
 >50K      7841
Name: income, dtype: int64

7841 / len(dataset), 24720 / len(dataset)

(0.2408095574460244, 0.7591904425539756)

split = StratifiedShuffleSplit(test_size=0.0030711587481956942)
for x, y in split.split(dataset, dataset['income']):
  df_x = dataset.iloc[x]
  df_y = dataset.iloc[y]

df_x.shape, df_y.shape

((32461, 16), (100, 16))

df_y.head()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income grupo
16531 48 Private 119722 Some-college 10 Married-civ-spouse Sales Husband Black Male 0 0 8 United-States <=50K 165
26676 29 Private 84366 10th 6 Married-spouse-absent Adm-clerical Unmarried White Female 0 0 40 Mexico <=50K 266
3124 19 Private 181546 Some-college 10 Never-married Adm-clerical Own-child White Female 0 0 40 United-States <=50K 31
27710 52 Private 317032 HS-grad 9 Married-civ-spouse Craft-repair Husband White Male 0 0 50 United-States >50K 277
23764 48 Private 195491 HS-grad 9 Divorced Sales Not-in-family White Female 0 0 30 United-States <=50K 237 
df_y['income'].value_counts()

 <=50K    76
 >50K     24
Name: income, dtype: int64

def amostragem_estratificada(dataset, percentual):
  split = StratifiedShuffleSplit(test_size=percentual, random_state=1)
  for _, y in split.split(dataset, dataset['income']):
    df_y = dataset.iloc[y]
  return df_y

df_amostra_estratificada = amostragem_estratificada(dataset, 0.003)
df_amostra_estratificada.shape

(98, 16)

3.5. Amostragem de Reservatório

A amostragem de reservatório é um método de amostragem usado principalmente quando o tamanho da população é desconhecido ou muito grande para ser acessado diretamente. Esse método permite que amostras aleatórias sejam retiradas de uma sequência de elementos, mesmo quando a população total não é conhecida antecipadamente.

3.5.1. Como funciona a amostragem de reservatório:

  1. Inicialização:

    • O processo começa com a seleção de um número fixo de elementos, igual ao tamanho da amostra desejada. Esses elementos são armazenados no “reservatório” (ou seja, um conjunto de amostras iniciais).

    • Se o número total de elementos a serem amostrados for N e o tamanho da amostra desejada for k, então os primeiros k elementos da sequência são diretamente colocados no reservatório.

  2. Seleção de elementos subsequentes:

    • A cada novo elemento que chega após os primeiros k elementos, o elemento é considerado para entrar no reservatório com uma probabilidade de k / n, onde n é o número total de elementos processados até o momento.

    • Se o novo elemento for selecionado, ele substitui um dos elementos existentes no reservatório de forma aleatória. Caso contrário, o elemento é descartado.

  3. Finalização:

    • O processo continua até que todos os elementos tenham sido processados ou até que se tenha percorrido toda a população ou sequência de dados.

    • Ao final, o reservatório contém uma amostra aleatória de k elementos da sequência original.

3.5.2. Exemplo de amostragem de reservatório:

Imagine que você está processando uma sequência de dados com 1.000.000 de elementos e deseja selecionar uma amostra de 10. Para isso, você faria o seguinte:

  1. Inicialmente, você seleciona os primeiros 10 elementos e os armazena no reservatório.

  2. Para o 11º elemento, você o seleciona com uma probabilidade de 10/11 e, caso selecionado, substitui um dos 10 elementos existentes no reservatório.

  3. Esse processo continua até o último elemento da sequência. No final, você terá uma amostra aleatória de 10 elementos da sequência total.

3.5.3. Vantagens da amostragem de reservatório:

  • Eficiência em grandes populações: A amostragem de reservatório é especialmente útil quando o tamanho total da população ou sequência é muito grande ou desconhecido, pois você não precisa armazenar todos os elementos, apenas a amostra selecionada.

  • Simples de implementar: O algoritmo de amostragem de reservatório é relativamente simples e não requer muita memória, tornando-o adequado para grandes fluxos de dados em tempo real.

  • Amostragem aleatória uniforme: Esse método garante que todos os elementos da sequência têm a mesma probabilidade de serem selecionados na amostra.

3.5.4. Desvantagens da amostragem de reservatório:

  • Desperdício de elementos não selecionados: Embora o método garanta que a amostra seja aleatória, ele pode resultar em perda de informações sobre elementos que não foram selecionados, já que a amostra não reflete a sequência inteira.

  • Seleção não ideal em casos de grandes variabilidades: Em populações muito desiguais ou com grande variabilidade entre os elementos, a amostragem de reservatório pode não ser tão eficaz quanto outros métodos de amostragem estratificada ou sistemática.

3.5.5. Aplicações:

A amostragem de reservatório é comumente usada em:

  • Processamento de fluxos de dados em tempo real: Por exemplo, na análise de logs de servidores ou dados de sensores, onde a população pode ser muito grande ou infinita.

  • Sistemas de recomendação e aprendizado de máquina: Quando se trabalha com grandes conjuntos de dados e é necessário extrair uma amostra representativa sem armazenar toda a população.

  • Pesquisas em que a população é grande e o acesso direto a todos os dados não é viável.

3.5.6. Exemplo em código

def amostragem_reservatorio(sequencia, tamanho_amostra):
    reservatorio = []
    for i, elemento in enumerate(sequencia):
        if i < tamanho_amostra:
            # Adiciona os primeiros 'tamanho_amostra' elementos diretamente ao reservatório
            reservatorio.append(elemento)
        else:
            # Para os elementos subsequentes, substitui aleatoriamente um elemento do reservatório
            if random.random() < tamanho_amostra / (i + 1):
                substituido = random.randint(0, tamanho_amostra - 1)
                reservatorio[substituido] = elemento
    return reservatorio

# Exemplo de uso:
sequencia = range(1, 100001)  # Uma sequência de 100.000 elementos
tamanho_amostra = 10

amostra = amostragem_reservatorio(sequencia, tamanho_amostra)
print(amostra)

[73600, 73714, 56813, 52272, 59438, 55232, 25164, 45053, 89981, 94223]

3.5.7. Exemplo usando o dataset inicial

stream = list(range(len(dataset)))
print(stream)

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1032, 1033, 1034, 1035, 1036, 1037, 1038, 1039, 1040, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1048, 1049, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 1072, 1073, 1074, 1075, 1076, 1077, 1078, 1079, 1080, 1081, 1082, 1083, 1084, 1085, 1086, 1087, 1088, 1089, 1090, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1098, 1099, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1107, 1108, 1109, 1110, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1140, 1141, 1142, 1143, 1144, 1145, 1146, 1147, 1148, 1149, 1150, 1151, 1152, 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168, 1169, 1170, 1171, 1172, 1173, 1174, 1175, 1176, 1177, 1178, 1179, 1180, 1181, 1182, 1183, 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2180, 2181, 2182, 2183, 2184, 2185, 2186, 2187, 2188, 2189, 2190, 2191, 2192, 2193, 2194, 2195, 2196, 2197, 2198, 2199, 2200, 2201, 2202, 2203, 2204, 2205, 2206, 2207, 2208, 2209, 2210, 2211, 2212, 2213, 2214, 2215, 2216, 2217, 2218, 2219, 2220, 2221, 2222, 2223, 2224, 2225, 2226, 2227, 2228, 2229, 2230, 2231, 2232, 2233, 2234, 2235, 2236, 2237, 2238, 2239, 2240, 2241, 2242, 2243, 2244, 2245, 2246, 2247, 2248, 2249, 2250, 2251, 2252, 2253, 2254, 2255, 2256, 2257, 2258, 2259, 2260, 2261, 2262, 2263, 2264, 2265, 2266, 2267, 2268, 2269, 2270, 2271, 2272, 2273, 2274, 2275, 2276, 2277, 2278, 2279, 2280, 2281, 2282, 2283, 2284, 2285, 2286, 2287, 2288, 2289, 2290, 2291, 2292, 2293, 2294, 2295, 2296, 2297, 2298, 2299, 2300, 2301, 2302, 2303, 2304, 2305, 2306, 2307, 2308, 2309, 2310, 2311, 2312, 2313, 2314, 2315, 2316, 2317, 2318, 2319, 2320, 2321, 2322, 2323, 2324, 2325, 2326, 2327, 2328, 2329, 2330, 2331, 2332, 2333, 2334, 2335, 2336, 2337, 2338, 2339, 2340, 2341, 2342, 2343, 2344, 2345, 2346, 2347, 2348, 2349, 2350, 2351, 2352, 2353, 2354, 2355, 2356, 2357, 2358, 2359, 2360, 2361, 2362, 2363, 2364, 2365, 2366, 2367, 2368, 2369, 2370, 2371, 2372, 2373, 2374, 2375, 2376, 2377, 2378, 2379, 2380, 2381, 2382, 2383, 2384, 2385, 2386, 2387, 2388, 2389, 2390, 2391, 2392, 2393, 2394, 2395, 2396, 2397, 2398, 2399, 2400, 2401, 2402, 2403, 2404, 2405, 2406, 2407, 2408, 2409, 2410, 2411, 2412, 2413, 2414, 2415, 2416, 2417, 2418, 2419, 2420, 2421, 2422, 2423, 2424, 2425, 2426, 2427, 2428, 2429, 2430, 2431, 2432, 2433, 2434, 2435, 2436, 2437, 2438, 2439, 2440, 2441, 2442, 2443, 2444, 2445, 2446, 2447, 2448, 2449, 2450, 2451, 2452, 2453, 2454, 2455, 2456, 2457, 2458, 2459, 2460, 2461, 2462, 2463, 2464, 2465, 2466, 2467, 2468, 2469, 2470, 2471, 2472, 2473, 2474, 2475, 2476, 2477, 2478, 2479, 2480, 2481, 2482, 2483, 2484, 2485, 2486, 2487, 2488, 2489, 2490, 2491, 2492, 2493, 2494, 2495, 2496, 2497, 2498, 2499, 2500, 2501, 2502, 2503, 2504, 2505, 2506, 2507, 2508, 2509, 2510, 2511, 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31863, 31864, 31865, 31866, 31867, 31868, 31869, 31870, 31871, 31872, 31873, 31874, 31875, 31876, 31877, 31878, 31879, 31880, 31881, 31882, 31883, 31884, 31885, 31886, 31887, 31888, 31889, 31890, 31891, 31892, 31893, 31894, 31895, 31896, 31897, 31898, 31899, 31900, 31901, 31902, 31903, 31904, 31905, 31906, 31907, 31908, 31909, 31910, 31911, 31912, 31913, 31914, 31915, 31916, 31917, 31918, 31919, 31920, 31921, 31922, 31923, 31924, 31925, 31926, 31927, 31928, 31929, 31930, 31931, 31932, 31933, 31934, 31935, 31936, 31937, 31938, 31939, 31940, 31941, 31942, 31943, 31944, 31945, 31946, 31947, 31948, 31949, 31950, 31951, 31952, 31953, 31954, 31955, 31956, 31957, 31958, 31959, 31960, 31961, 31962, 31963, 31964, 31965, 31966, 31967, 31968, 31969, 31970, 31971, 31972, 31973, 31974, 31975, 31976, 31977, 31978, 31979, 31980, 31981, 31982, 31983, 31984, 31985, 31986, 31987, 31988, 31989, 31990, 31991, 31992, 31993, 31994, 31995, 31996, 31997, 31998, 31999, 32000, 32001, 32002, 32003, 32004, 32005, 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def amostragem_reservatorio(dataset, amostras):
    # Criar o stream diretamente como uma lista de índices
    stream = list(range(len(dataset)))

    # Inicializa o reservatório com os primeiros 'amostras' índices do stream
    reservatorio = stream[:amostras]

    # Realiza a amostragem de reservatório
    for i in range(amostras, len(dataset)):
        j = random.randrange(i + 1)
        if j < amostras:
            reservatorio[j] = stream[i]

    return dataset.iloc[reservatorio]

df_amostragem_reservatorio = amostragem_reservatorio(dataset, 100)
df_amostragem_reservatorio.shape

(100, 16)

df_amostragem_reservatorio.head()
age workclass final-weight education education-num marital-status occupation relationship race sex capital-gain capital-loos hour-per-week native-country income grupo
3345 45 Private 125194 11th 7 Never-married Machine-op-inspct Not-in-family Black Female 0 0 40 United-States <=50K 33
1558 25 Private 352057 HS-grad 9 Never-married Transport-moving Not-in-family White Male 0 0 65 United-States <=50K 15
16441 26 Private 485117 Assoc-acdm 12 Never-married Craft-repair Not-in-family White Male 0 0 40 United-States <=50K 164
29755 60 Federal-gov 119832 Some-college 10 Married-civ-spouse Adm-clerical Husband White Male 7298 0 40 United-States >50K 297
15300 54 Private 192862 HS-grad 9 Divorced Exec-managerial Unmarried White Female 0 0 45 United-States <=50K 153

3.6. Comparativo dos resultados

# Criação do DataFrame de comparação
df_comparacao = pd.DataFrame({
    'Método': [
        'Original (dataset)',
        'Amostra Aleatória Simples',
        'Amostra Sistemática',
        'Amostra por Agrupamento',
        'Amostra Estratificada',
        'Amostragem por Reservatório'
    ],
    'Média da Idade': [
        dataset['age'].mean(),
        df_amostra_aleatoria_simples['age'].mean(),
        df_amostra_sistematica['age'].mean(),
        df_amostra_agrupamento['age'].mean(),
        df_amostra_estratificada['age'].mean(),
        df_amostragem_reservatorio['age'].mean()
    ]
})

df_comparacao
Método Média da Idade
0 Original (dataset) 38.581647
1 Amostra Aleatória Simples 39.410000
2 Amostra Sistemática 37.570000
3 Amostra por Agrupamento 38.970000
4 Amostra Estratificada 37.010204
5 Amostragem por Reservatório 39.590000

3.7. Sobre a comparação

  1. Original (dataset):

    • A média de idade do dataset original é 38.58.

    • Este valor serve como referência para a comparação com as amostras extraídas pelas diferentes técnicas de amostragem.

  2. Amostra Aleatória Simples:

    • A média de idade é 39.41, que é ligeiramente superior à média do dataset original.

    • Isso pode ser devido à natureza aleatória dessa amostra, o que pode ter levado à seleção de um subconjunto com uma média de idade maior do que a do conjunto original.

  3. Amostra Sistemática:

    • A média de idade é 37.57, que é ligeiramente inferior à média do dataset original.

    • A amostragem sistemática tende a selecionar os dados de uma forma regular, o que pode não capturar adequadamente a variação na idade presente no conjunto completo.

  4. Amostra por Agrupamento:

    • A média de idade é 38.97, que é bastante próxima à média do dataset original (38.58).

    • Isso sugere que a amostragem por agrupamento foi eficaz em manter uma média de idade similar ao conjunto original, o que é esperado, pois o agrupamento divide os dados em subconjuntos que são representativos da população total.

  5. Amostra Estratificada:

    • A média de idade é 37.01, o valor mais baixo entre as amostras analisadas.

    • A amostragem estratificada divide a população em estratos, e pode ser que a distribuição das idades nos estratos tenha causado uma média mais baixa. Isso pode ocorrer se a amostra estratificada tiver sido composta predominantemente por estratos com idades mais baixas.

  6. Amostragem por Reservatório:

    • A média de idade é 39.59, a mais alta entre todas as amostras.

    • Isso sugere que a amostragem por reservatório pode ter dado mais ênfase à seleção de indivíduos com idades mais altas, o que pode ser o resultado do processo de amostragem aleatória com substituição que favorece a escolha de elementos de idades maiores ao longo do tempo.

[BBG20] (1,2,3)

Peter Bruce, Andrew Bruce, and Peter Gedeck. Practical Statistics for Data Scientists. O’Reilly Media, Inc.,, 2 edition, 2020. URL: https://learning.oreilly.com/library/view/practical-statistics-for/9781492072935/.